Optimal. Leaf size=119 \[ -\frac {i (c-i d)^2 (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m}+\frac {2 c d (a+i a \tan (e+f x))^m}{f m}-\frac {i d^2 (a+i a \tan (e+f x))^{m+1}}{a f (m+1)} \]
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Rubi [A] time = 0.17, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3543, 3527, 3481, 68} \[ -\frac {i (c-i d)^2 (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m}+\frac {2 c d (a+i a \tan (e+f x))^m}{f m}-\frac {i d^2 (a+i a \tan (e+f x))^{m+1}}{a f (m+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3481
Rule 3527
Rule 3543
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx &=-\frac {i d^2 (a+i a \tan (e+f x))^{1+m}}{a f (1+m)}+\int (a+i a \tan (e+f x))^m \left (c^2-d^2+2 c d \tan (e+f x)\right ) \, dx\\ &=\frac {2 c d (a+i a \tan (e+f x))^m}{f m}-\frac {i d^2 (a+i a \tan (e+f x))^{1+m}}{a f (1+m)}+(c-i d)^2 \int (a+i a \tan (e+f x))^m \, dx\\ &=\frac {2 c d (a+i a \tan (e+f x))^m}{f m}-\frac {i d^2 (a+i a \tan (e+f x))^{1+m}}{a f (1+m)}-\frac {\left (i a (c-i d)^2\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {2 c d (a+i a \tan (e+f x))^m}{f m}-\frac {i (c-i d)^2 \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 f m}-\frac {i d^2 (a+i a \tan (e+f x))^{1+m}}{a f (1+m)}\\ \end {align*}
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Mathematica [B] time = 17.67, size = 246, normalized size = 2.07 \[ -\frac {i 2^{m-1} \left (e^{i f x}\right )^m e^{-i (e m+e+2 f m x+f x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{m+1} \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left (\frac {2 \left (c^2+d^2\right ) e^{i (e (m+2)+2 f (m+1) x)}}{m+1}+\frac {(c-i d)^2 e^{i (e (m+4)+2 f (m+2) x)} \, _2F_1\left (1,1;m+3;-e^{2 i (e+f x)}\right )}{m+2}+\frac {(c+i d)^2 e^{i m (e+2 f x)} \left (e^{2 i (e+f x)}+m+1\right )}{m (m+1)}\right )}{f} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} + 2 i \, c d - d^{2} + {\left (c^{2} - 2 i \, c d - d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{2} + d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m}}{e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.25, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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